Curves with few bad primes over cyclotomic $\mathbb{Z}_\ell$-extensions

Abstract: Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$, and write $\mathcal{O}S^\times$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$. Now instead of a number field, let $K=\mathbb{Q}{\infty,\ell}$ which denotes the $\mathbb{Z}_\ell$-cyclotomic extension of $\mathbb{Q}$. We show that the $S$-unit equation $\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$, has infinitely many solutions for $\ell \in {2,3,5,7}$, where $S$ consists only of the totally ramified prime above $\ell$. Moreover, for every prime $\ell$, we construct infinitely many elliptic or hyperelliptic curves defined over $K$ with good reduction away from $2$ and $\ell$. For certain primes $\ell$ we show that the Jacobians of these curves in fact belong to infinitely many distinct isogeny classes.